Stochastic Galerkin Methods without Uniform Ellipticity Marcus - PowerPoint PPT Presentation

Stochastic Galerkin Methods without Uniform Ellipticity Marcus Sarkis (WPI/IMPA) Collaborator: Juan Galvis (Texas A & M) WPI/IMPA RICAM MS & AEE-Workshop4, Dec13/2011 Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 1 / 28

Problem of interest Consider the Darcy’s equation −∇ x . ( κ ( x , ω ) ∇ x u ( x , ω )) = f ( x , ω ) , for x ∈ D ⊂ R d   u ( x , ω ) = 0 , on ∂ D  κ ( x , ω ) = e W ( x ,ω ) ◮ W ( x , ω ) = � ∞ k =1 a k ( x ) ξ k ( ω ) ◮ ξ k are iid standard normal random variables ◮ e W ( x ,ω ) ∈ (0 , ∞ ) not bounded, not uniformly elliptic f ( x , ω ) ◮ Random forcing term Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 2 / 28

Outline One-dimensional log-normal noise White noise framework Countable infinite-dimensional log-normal noise Galerkin spectral method Discretization, well-posedness, a priori error Numerical results Conclusions Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 3 / 28

Breeding analysis on Log-Normal without ellipticity November 2005: I. Babuska, F. Nobile and R. Tempone: A stochastic collocation method for elliptic partial differential equations with random input data. March 2008: J. Galvis and S., Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. March 2009: X. Wan, B. Rozovskii and G. E. Karniadakis, A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. May 2009: C.J. Gittelson, Stochastic Galerkin discretization of the lognormal isotropic diffusion problem. June 2010: J. Charrier, Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. January 2011: A. Mugler and H.-J. Starkloff, On elliptic partial differential equations with random coefficients. Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 4 / 28

References Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity, Juan Galvis and Marcus Sarkis. SIAM J. Numer. Anal., Vol. 47(5), pp. 3624-3651, 2009. Regularity results for the ordinary product stochastic pressure equation, Juan Galvis and Marcus Sarkis. Submitted. Preprint serie IMPA A 692, 2011. An introduction to infinite-dimensional analysis, Giuseppe Da Prato. Universitext, Springer-Verlag, Berlin, 2006. Stochastic analysis, Ichiro Shigekawa. Translations of Mathematical Monographs, Vol. 224, AMS, Providence, RI, 2004. Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 5 / 28

A simple example: one-dimensional log-normal noise − ( e a 1 ( x ) ξ 1 u x ) x = f ( x ) , in D = (0 , 1) Young Inequality: Let C 1 := max x ∈ D | a 1 ( x ) | and ǫ > 0 C 2 C 2 2 ǫ e − ǫ 2 ξ 2 ǫ 2 ξ 2 1 1 ≤ e a 1 ( x ) ξ 1 ≤ e 1 e − 2 ǫ e 1 Idea: Use weights of the type e s ξ 2 1 (easy to integrate) � ∞ � 1 G ( y ) e sy 2 − 1 G ( ξ 1 ) e s ξ 2 2 y 2 dy 1 d ξ 1 = √ 2 π −∞ Lax Milgram: C 2 1 H − 1 (0 , 1) e ǫξ 2 � u x ( · , ξ 1 ) � 2 L 2 (0 , 1) ≤ C 2 ǫ � f � 2 P e 1 Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 6 / 28

A simple example: one-dimensional log-normal noise C 2 1 H − 1 ( D ) e ǫξ 2 � u x ( · , ξ 1 ) � 2 L 2 ( D ) ≤ C 2 ǫ � f � 2 P e 1 Integrate over ξ 1 , (0 < ǫ < 1 / 2) u ∈ H 1 0 ( D ) ⊗ ( L 2 )( d ξ 1 ) bounded by f ∈ H − 1 ( D ) ⊗ ( L 2 ) ǫ ( d ξ 1 ) Integrate over ξ 1 , ( ǫ > 0 and s ∈ R such that s + ǫ < 1 / 2) � ∞ � 1 L 2 ( D ) e sy 2 − 1 L 2 ( D ) e s ξ 2 2 y 2 dy || u x ( · , ξ 1 ) || 2 � u x ( · , y ) � 2 √ 1 d ξ 1 = 2 π −∞ � ∞ C 2 1 e ( s + ǫ ) y 2 − 1 1 2 y 2 dy C 2 ǫ � f � 2 ≤ √ P e H − 1 2 π −∞ C 2 ǫ (1 − 2( s + ǫ )) − 1 1 C 2 2 � f � 2 = P e H − 1 ( D ) For s � = 0, u x ∈ L 2 ( D ) ⊗ L 2 s ( d ξ 1 ) important when f ( x , ξ 1 ) Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 7 / 28

A simple example: one-dimensional log-normal noise RHS f ( x , ξ 1 ) with lack ( s < 0) extra ( s > 0 ) decay in ξ 1 C 2 � � L 2 ( D ) e s ξ 2 1 H − 1 ( D ) e ( s + ǫ ) ξ 2 � u x ( · , ξ 1 ) � 2 1 d ξ 1 ≤ C 2 � f ( · , ξ 1 ) � 2 1 d ξ 1 P e ǫ Note: u → s , f → s + ǫ , and v → − ( s + ǫ ) Given f ∈ H − 1 ( D ) ⊗ L 2 s + ǫ ( d ξ 1 ), find u ∈ H 1 0 ( D ) ⊗ L 2 s ( d ξ 1 ) such that ∀ v ∈ H 1 0 ( D ) ⊗ L 2 a ( u , v ) = f ( v ) , − ( s + ǫ ) ( d ξ 1 ) Existence and uniqueness (inf-sup condition) Galvis and S. (09’) Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 8 / 28

We need a theoretical framework to deal: Infinite-dimensional case (Gaussian measure) Generalized Wiener-chaos expansions (on Hilbert spaces) Galerkin spectral methods (Explicit computations) A priori error estimates (Natural to establish) Regularity theory (Derivatives in ω and x ) Gaussian Sobolev (Hilbert) spaces Constants that depend on few quantities Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 9 / 28

Probability space: constructed from a pair ( H , A ) Using KL: ◮ H := L 2 ( D ) ◮ Correlation operator: C v ( x ) := � D k ( x , ˜ x ) v (˜ x ) d ˜ x ◮ C → : Eigenfunctions q k , eigenvalues µ k ◮ A = C − 1 ◮ A → : Eigenfunctions q k , eigenvalues λ k = 1 /µ k Using convolution (1D- Smoothed white noise) ◮ H := L 2 ( R ) dx 2 + x 2 + 1 ◮ A := − d 2 ◮ A → : Hermite functions q k , eigenvalues λ k = 2 k Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 10 / 28

White noise framework Hilbert space H , operator A , and H -orthonormal basis { q k } ∞ k =1 : ◮ Aq k = λ k q k , k = 1 , 2 , . . . ◮ 1 < λ 1 ≤ λ 2 ≤ · · · k =1 λ − 2 θ ◮ � ∞ < ∞ for some constant θ > 0 k H = � ∞ k =1 λ 2 p p ≥ 0, ξ ∈ H , � ξ � 2 p := � A p ξ � 2 k ( ξ, q k ) 2 H S ′ S p := { ξ ∈ H ; � ξ � p < ∞} S := ∩ p ≥ 0 S p Probability measure µ (Bochner-Minlos theorem) characterized by � E µ e i �· ,ξ � := S ′ e i � ω,ξ � d µ ( ω ) = e − 1 2 � ξ � 2 H , for all ξ ∈ S Probability space (Ω , F , P ) = ( S ′ , B ( S ′ ) , µ ) Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 11 / 28

Remarks: Fernique and change of variables µ ( S − θ ) = 1 where S − θ = { ω ∈ S ′ : � ω � − θ < ∞} � E µ e i �· ,ξ � := e i � ω,ξ � d µ ( ω ) = e − 1 2 � ξ � 2 H for all ξ ∈ S S − θ ( S − θ , B ( S − θ ) , µ ): normally distributed RV X ξ ( ω ) = � ω, ξ � Equivalent formulation: � µ ( h ) = e − 1 2 � ξ � 2 µ e i �· ,ξ � := A − 2 θ for all ξ ∈ S e i � h ,ξ � d ˜ E ˜ H µ Gaussian measure with covariance A − 2 θ ˜ µ ): normally distributed RV Y ξ ( h ) = � h , A θ ξ � ( H , B ( H ) , ˜ See Da Prato (06’) for θ = 1 / 2 and Galvis and S. (11’) Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 12 / 28

Gaussian Field W ( x , ω ) = � ω, φ x � In KL: − 1 ◮ φ x (˜ x ) = � ∞ x ) ≡ φ ( x , ˜ k =1 λ 2 q k ( x ) q k (˜ x ) k ◮ W ( x , ω ) = � ω, φ x � = � ∞ k =1 a k ( x ) � ω, q k � − 1 ◮ a k ( x ) = λ 2 q k ( x ) k ◮ ξ k ( ω ) = � ω, q k � i.i. normally distributed In convolution: ◮ φ x (˜ x ) ≡ φ ( x − ˜ x ) (smoothed window) ◮ W ( x , ω ) = � ω, φ x � = � ∞ k =1 a k ( x ) � ω, q k � ◮ a k ( x ) = � R φ ( x − ˜ x ) q k (˜ x ) d ˜ x ◮ ξ k ( ω ) = � ω, q k � i.i. normally distributed Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 13 / 28

Countable independent normals W ( x , ω ) = � ω, φ x � = � ∞ k =1 a k ( x ) ξ k ( ω ) − ǫλ − 2 θ + ǫλ − 2 θ − λ 2 θ k a 2 ξ k ( ω ) 2 ≤ a k ( x ) ξ k ( ω ) ≤ λ 2 θ k a 2 ξ k ( ω ) 2 k ( x ) k ( x ) k k 2 ǫ 2 2 ǫ 2 − ǫ � ω � 2 + ǫ � ω � 2 −||| φ ||| 2 ≤ � ω, φ x � ≤ ||| φ ||| 2 − θ − θ θ θ 2 ǫ 2 2 ǫ 2 ||| φ ||| 2 θ := sup x ∈ D � φ x � 2 θ ∞ ∞ � φ x � 2 � λ 2 θ k ( φ x , q k ) 2 � λ 2 θ k a 2 θ = H = k ( x ) < ∞ k =1 k =1 − θ := � ∞ � ω � 2 k =1 λ − 2 θ � ω, q k � 2 k ∞ � S ′ � ω � 2 � λ − 2 θ − θ d µ ( ω ) = < ∞ k k =1 Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 14 / 28

Two examples KL: √ µ k < ∞ ◮ � ∞ k =1 ◮ sup k � q k � L ∞ ( D ) < ∞ ◮ Take θ = 1 / 4, both conditions are satisfied Smoothed white noise: ◮ λ k = 2 k (note that � ∞ k =1 λ − 1 = ∞ ) k ◮ φ ( x − ˜ x ) a smooth window (the a k ( x ) decay fast) ◮ Take θ > 1, both conditions are satisfied Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) () SPDE-GALERKIN RICAM 2011 15 / 28